v2009.01.01 - Convex Optimization
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v2009.01.01 - Convex Optimization

v2009.01.01 - Convex Optimization v2009.01.01 - Convex Optimization

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320 CHAPTER 4. SEMIDEFINITE PROGRAMMING Literature on the max cut problem is vast because this problem has elegant primal and dual formulation, its solution is very difficult, and there exist many commercial applications; e.g., semiconductor design [104], quantum computing [338]. Our purpose here is to demonstrate how iteration of two simple convex problems can quickly converge to an optimal solution of the max cut problem with a 98% success rate, on average. 4.33 max cut is stated: ∑ maximize a ij (1 − x i x j ) 1 x 2 1≤i

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 321 Because an estimate of upper bound to max cut is needed to ascertain convergence when vector x has large dimension, we digress to derive the dual problem: Directly from (733), the Lagrangian is [53,5.1.5] (1321) L(x, ν) = 1 4 〈xxT , δ(A1) − A〉 + 〈ν , δ(xx T ) − 1〉 = 1 4 〈xxT , δ(A1) − A〉 + 〈δ(ν), xx T 〉 − 〈ν , 1〉 = 1 4 〈xxT , δ(A1 + 4ν) − A〉 − 〈ν , 1〉 (734) where quadratic x T (δ(A1+ 4ν)−A)x has supremum 0 if δ(A1+ 4ν)−A is negative semidefinite, and has supremum ∞ otherwise. The finite supremum of dual function g(ν) = sup L(x, ν) = x { −〈ν , 1〉 , A − δ(A1 + 4ν) ≽ 0 ∞ otherwise (735) is chosen to be the objective of minimization to dual (convex) problem minimize −ν T 1 ν subject to A − δ(A1 + 4ν) ≽ 0 (736) whose optimal value provides a least upper bound to max cut, but is not tight ( 1 4 〈xxT , δ(A1)−A〉< g(ν) , duality gap is nonzero). [132] In fact, we find that the bound’s variance with problem instance is too large to be useful for this problem; thus ending our digression. 4.34 To transform max cut to its convex equivalent, first define then max cut (733) becomes X = xx T ∈ S n (741) maximize 1 X∈ S n 4 〈X , δ(A1) − A〉 subject to δ(X) = 1 (X ≽ 0) rankX = 1 (737) 4.34 Taking the dual of dual problem (736) would provide (737) but without the rank constraint. [125]

320 CHAPTER 4. SEMIDEFINITE PROGRAMMING<br />

Literature on the max cut problem is vast because this problem has elegant<br />

primal and dual formulation, its solution is very difficult, and there exist<br />

many commercial applications; e.g., semiconductor design [104], quantum<br />

computing [338].<br />

Our purpose here is to demonstrate how iteration of two simple convex<br />

problems can quickly converge to an optimal solution of the max cut<br />

problem with a 98% success rate, on average. 4.33 max cut is stated:<br />

∑<br />

maximize a ij (1 − x i x j ) 1<br />

x<br />

2<br />

1≤i

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